The Minimum Spanning Forest with Balance Constraints ...

Minimum Spanning Forest with Balance Constraints - Herzterg, Poggi, Vidal

The Minimum Spanning Forestwith Balance Constraints Problem (MSFBC)

Ian Herzterg, Marcus Poggi, Thibault VidalDepartamento de Informatica, PUC-Rio, Brazil{iherzterg, poggi, vidalt} @inf.puc-rio.br

July 15, 2015

Outlines

1 Problem Definition and Complexity

2 Formulations

3 Solving MSFBCDual and Primal HeuristicsExact Methods

4 Experiments

5 An Application

6 Future Work

Problem Definition and Complexity

Minimum Spanning Forest with Balance Constraints

Instance:

a vertex set R integer weights pv, v ∈ R;nonnegative costs between all pairs of vertices and from allvertices to terminal 0(border), let V = R ∪ {0};

Find a minimum cost edge set F such that all connected components are:

acyclic;

either balanced (∑

v∈comp

pv = 0)

or connected to the terminal.

Instance:

a vertex set R integer weights pv, v ∈ R;nonnegative costs between all pairs of vertices and from allvertices to terminal 0(border), let V = R ∪ {0};

Find a minimum cost edge set F such that all connected components are:

acyclic;

either balanced (∑

v∈comp

pv = 0)

or connected to the terminal.

Figure: Long - Matching

Figure: Long - MSFBC

Theorem

MSFBC is NP-Hard

Steiner Problem in Graphs (SPG): G = (V,E), T ⊂ V , we e ∈ E, findminimum weight edge set E′ connecting all vertices in T .

Reduction:

Let R be V where all non-terminal vertices are doubled |R| = 2.|V | − |T |

Choose a vertex r ∈ T and assign weights pv as follows:

pv = −1, ∀ v ∈ T \ {r}pr = |T | − 1For each pair v, w of doubled non-terminal vertex assignpv = +1, pw = −1

Connect all pairs of non-terminal vertices by a zero cost edge, and each ofthem to the other vertices as before (both when also non-terminal vertex).

Theorem

MSFBC is NP-Hard

Steiner Problem in Graphs (SPG): G = (V,E), T ⊂ V , we e ∈ E, findminimum weight edge set E′ connecting all vertices in T .

Reduction:

Let R be V where all non-terminal vertices are doubled |R| = 2.|V | − |T |

Choose a vertex r ∈ T and assign weights pv as follows:

pv = −1, ∀ v ∈ T \ {r}pr = |T | − 1For each pair v, w of doubled non-terminal vertex assignpv = +1, pw = −1

Connect all pairs of non-terminal vertices by a zero cost edge, and each ofthem to the other vertices as before (both when also non-terminal vertex).

Steiner Problem in Graphs to MSFBC

Optimal Solutions SPG and MSFBC

Alternative Optimal Solutions MSFBC (Decomposition for SPG ?)

Formulations

Undirected formulation for the MSFBC:

min∑e∈E

dexe s.t. (1)

∑e∈δ(S)

xe ≥ 1, ∀S ⊂ V, s.t.∑v∈S

pv 6= 0 (2)

xe ∈ {0, 1}, ∀e ∈ E, (3)

Formulations

Directed formulation for the MSFBC:

min∑e∈E

dexe s.t. (4)

∑a∈δ+(S)

xa ≥ 1, ∀S ⊂ R, s.t.∑v∈S

pv > 0 (5)

∑a∈δ−(S)

xa ≥ 1, ∀S ⊂ R, s.t.∑v∈S

pv < 0 (6)

xa + xa′ ≤ 1, ∀a = (i, j), a′ = (j, i) ∈ ER (7)

xa ∈ {0, 1},∀a ∈ E (8)

Formulations

Set Partitioning formulation (SPF) for the MSFBC:

Let J be the set of all subsets Rj of Rcj is the cost of the MST connecting subset Rj , plus thecheapest connection to the terminal if Rj is not balanced

min∑j∈J

cjxj s.t. (9)

∑j∈J

avjxj = 1, ∀v ∈ R (10)

xj ∈ {0, 1},∀j ∈ J, (11)

{1 if v ∈ Sj

0 if v /∈ Sj(12)

Solving MSFBC

Dual and Primal Heuristics

Dual Heuristics: Dual Ascent over Directed Formulation

Dual Ascent: Selects connected components until they becomebalancedSelection: Greedy and RandomReverse delete step to generate primal solution

Solving MSFBC

Dual Ascent method - Dual heuristic

input : A dual feasible solution πoutput: A maximal dual feasible solution π’

Initialization: Build Gπ = (V, E) from the saturated arcs in ππ′ ← π;while exists a violated cut W ∈ Gπ do

W ← selectViolatedCut();

if∑v∈W

pv > 0 then

Augment π′W until at least one arc in δ−(W) becomessaturated;

else if∑v∈W

pv < 0 then

Augment π′W until at least one arc in δ+(W) becomessaturated;

endAdd the newly saturated arcs in Gπ;

endreturn π′;

Solving MSFBC

Primal Heuristics - Iterated Local Search (ILS) (enumerativeneighbourhoods)

Executed 10 times for each instance, with 25 shakingoperations each.simple local improvement procedures (swap, relocate, break,merge, break1-insert1) for each pair of trees T1 and T2, inrandom order.Hybrid set covering formulation (under development).

Solving MSFBC

ILS - Primal Heuristic

input : The graph G = (V ,E1)output: A set of balanced trees S

Initialization: Generate an initial solution I by computing theminimum spanning tree for the graph G and disconnecting the setedges whose costs exceeds the average edge cost in the tree. Forevery unbalanced tree, connect it to its closest border point.Itshake ← 0;S ← I;S* ← S;while Itshake < ItMAX do

S ← LocalSearch(S);if c(S) < c(S*) then

S* ← S;Itshake ← 0;

endforeach ItSC iterations do S ← SetCovering(S) if Itssuccessive iterations without improvement then

S ← Shake(S) or S ← Shake(S*) with 50% chance each;Itshake++;Its ← 0;

endreturn S*;

Solving MSFBC

Exact Methods

Branch-and-Cut

Directed formulationUses primal bounds and dual bound to fix arcs by reduced cost

Linear Relaxation

Accelerates cut separation by testing whether connectcomponents considering all edges with current xa positive arebalanced

Keeps current graph and runs DFS

Solves Max-Cut for all pairs

Solving MSFBC

Exact Methods

Branch-and-Cut

Linear Relaxation

Solving MSFBC

Exact Methods

Branch-and-Cut

Linear Relaxation

Solving MSFBC

Exact Methods

input : The undirected graph G1 = (V ,E1) and the directedgraph G2 = (V ,E2), both described by the solution set x

output: The set of cuts S

Initialization: For each node v in V , assign visited[v] ← false. LetC be the list of connected components and S the list of cuts.foreach node v in V do

if !visited[v] thenRun a depth-first-search on graph G1, starting from nodev. Add the discovered connected component in C.

endforeach connected component ck ∈ C do

foreach pair of nodes (i,j) ∈ ck do{s’, MaxFlow} ← minCutMaxFlow(i,j,G2);if s’ is unbalanced, MaxFlow < 1 and s’ 6∈ S then

S ← S + s’end

endforeach node i ∈ ck and a single node j /∈ ck do

s’ ← minCutMaxFlow(i,j,G2);if s’ is unbalanced, MaxFlow < 1 and s’ 6∈ S then S ← S+ s’

endif s(ck) is unbalanced and s(ck) /∈ S then S ← S + s(ck)

endreturn S;

Solving MSFBC

Exact Methods

Solving Set Partitioning’s Linear Relaxation

Column Generation subproblem:

Minimum Tree with Profits on Vertices: too hard

Relaxation for the SPF:

Based on Uchoa et al.(2008)’s q − arborFind a minimum cost connected component with limitednumber of vertices and vertex degree (q − arbor)

Solving MSFBC

Exact Methods

Solving SPF’s relaxed Linear Relaxation

Let T (v, a, q) be the minimum cost of a q − arbor with root invertex v where v has arity d and the total q − arbor has qvertices.Let inCost(v, a, q) be the cost without connecting the to theterminalLet c2terminal(v, a, q) be the cost to connect to the terminal.Finally let p(v, a, q) be the balance of q − arbor (a, v, q).

Solving MSFBC

Exact Methods

Solving SPF’s relaxed Linear Relaxation

The dynamic programming recursion follows:

T (v, a, q) = minw,ar,q2

{inCost(v, a−1, q1)+inCost(w, ar, q2)+dvw+∆}

the minimum is taken over, w 6= v, all ar and with q1, q2 suchthat q1 + q2 = q;∆ is equal to zero, when p(v, a− 1, q1) + p(w, ar, q2) == 0and to the minimum of c2terminal(v, a, q1) andc2terminal(w, ar, q2), otherwise.

Solving MSFBC

Exact Methods

q − arbor example

Experiments

Instances

MSFBC - How good are we ?

Preliminary Tests on Random Instances

Equal number of positive and negative residuesReport on instances with up to 512 pairs of vertices

Experiments

An Application

Signal Processing: 2D Phase Unwrapping

Figure: Wrapping effect on a 2D phase image.(a) Absolute phase image(b) Wrapped phase image.

An Application

Signal Processing: 2D Phase Unwrapping

MSFBC as a new model for 2D Phase Unwrapping

Does a solution with a smaller value for the MSFBC implies a betterunwrapping ?

An Application

Figure: Head. (a) Goldstein (b) Matching (c) MSFBC

An Application

Figure: Long (a) Goldstein (b) Matching (c) MSFBC

An Application

Figure: Long (a) Goldstein (b) Matching (c) MSFBC

Future Work

Benchmark with tailored instances

Gaps appear to be way larger than for Steiner: polyhedral study

Column Generation formulation:

How good would be SPF bounds for transformed Steinerinstances?What would be an ng − route equivalent SPF relaxation?Would Subset Row Cuts be effective?Limited memory SRC’s

Future Work

Thanks!

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